Gambling: Understanding the odds in numbers

David Spiegelhalter

About the author

David Spiegelhalter is Winton Professor of the Public Understanding of Risk at the University of Cambridge, UK. His professional background is in medical statistics. He is a frequent contributor to the media, and has been an advisor on risk issues as diverse as breast implants, volcanic ash, clinical trials, surgical mortality, and the inquiry into mass murderer Harold Shipman. He is a Fellow of the Royal Society and was awarded an OBE in 2006. You can see more of David’s explorations of risk at the Understanding Uncertainty website, or follow him on Twitter at @undunc

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Is there such a thing as a lucky person or a lucky streak? And does belief in good and bad luck play a part in whether we are prepared to take chances?

“The next best thing to gambling and winning is gambling and losing.” So said legendary gambler Nick “The Greek” Dandalos, and he would know. Over the course of his playing career he’s thought to have won and lost $500 million.

Gamblers and statisticians share a love of odds, though the stakes on the table couldn’t be more different between the two. But us statisticians are allowed to fantasise a little, so here’s a hypothetical challenge. Let’s say you fancied a Maserati costing $100,000, but sadly you only had a dollar, what form of gambling will give you the best chance of getting the sports car? I’ll even make it a little easier for you by giving you three choices – a punt on the lottery, a flutter on the horses or a spin on the roulette wheel.

While you think about this, you might be surprised to discover how far back our urge for gambling stretches. Devices that embody randomness have been used in leisure for at least 5,000 years. The ancient Egyptians, presumably when nothing good was on television, sat around and played board games in which moves were decided by throwing an astragalus, which is a bone in the heel that can land on one of four sides.

Gambling became so popular that the Romans tried to restrict it to Saturdays. Even the Emperor Claudius played obsessively and wrote a book called How to win at dice. People continued for centuries to make bets and quote odds: in Paris in 1588 you could get 5 to 1 against the Spanish Armada sailing to invade England, although this was probably a ruse by the Spanish.

Calculated risk

What is remarkable is that in all this time, right up until the Renaissance, nobody analysed gambles mathematically. This could be partly because people thought the outcomes were decided by some external force of fate, but it is now thought that the gap between theory and practice was too great, and therefore no-one had the idea of putting a number on chance - in other words calculate probability.

Not until the Chevalier de Mere, it seems. The Chevalier was a very perceptive gambler in Paris in the 1650s, and he reckoned that odds were slightly in his favour of getting a six in four throws of a single dice, but that the odds were slightly against him of throwing two dice and getting a double 6 in 24 throws. By chance (or fate) he brought the problem to two of the smartest mathematicians at that or any other time, Blaise Pascal and Pierre de Fermat. They confirmed that he had a 52% chance of winning the first bet, and a 49% chance of winning the second, so the Chevalier had got it right from what must have been very extensive and expensive experience.

The Chevalier also presented the mathematicians with the ‘problem of points’ – if a game has to stop before the end, in what proportions should the stake be divided up? A modern counterpart is the Duckworth-Lewis method in cricket, used for allocating runs scored by a team when a match is stopped by the weather – which in England is pretty much all the time. If you thought the game of cricket is difficult to understand, the Duckworth-Lewis method is almost totally incomprehensible to even the most ardent fan. No surprise, then, to hear that statisticians designed it.

After the Chevalier’s time the scientific assessment of odds disappeared for a few centuries, as the 1700s became a golden age of betting on the basis of gut-feelings rather than calculations. It was the time of ‘eccentric wagers’, such as a large sum won by the Count de Buckeburg in 1735 for riding from London to York seated backwards on a horse.

There were also huge bets on cricket matches, and a predictable consequence was the match-fixing scandals that erupted in the early 1800s, with spectators at Lord’s Cricket Ground (the so-called home of the game) bewildered at the sight of two nobbled teams both desperately trying to lose. A reaction followed, with bookmakers being banned from Lord’s in 1817, lotteries banned in 1826, and the Gaming Act of 1845 making gambling debts unenforceable by law. Cricket was turned into the archetypal gentlemen’s game. Until recently, that is. Heavy gambling has again brought match-fixing to the fore; and more worrying is the ability to bet on every minor detail of the match, known as spot-fixing, which has led to high-profile cases of players being bribed to mess up specific bowls.

Playing the odds

So back to the Maserati, and which of the three gambling options would give you the best odds. If you were to buy a single lottery ticket, and if your choice of six numbers matches the five winning balls plus the bonus number (a seventh ball drawn), then this generally wins around £100,000 in the UK (which for arguments sake let’s say is close enough to our dollar target – it will certainly cover the cost of the sports car). The probability of this happening is 1 in 2,330,636, which is about my chance of having a heart attack or stroke in the half-hour it takes for me to go to the shop for the ticket.

What if you were to go for an accumulator on the horses, that is, pick a race meeting with six races, and in each race choose a horse at medium odds of around 6 to 1 against? Then the accumulator, in which the winnings of each race are passed to the next horse, will give you 7×7×7×7×7×7 = $117,000 if they all win. Given a bookmakers margin of, say, 12% for each bet, the true odds may be around 1 in 230,000.

Which leaves the roulette wheel. If you can find a casino to let you bet $1, place it on your lucky number between 1 and 36. When it wins, either leave the $36 there or move it to another number. When that comes up too, move the $1,296 you now have to another number, or leave it where it is – it doesn’t make any difference to the odds, but somehow it seems that the chance increases when the money is moved. When that comes up you will have $46.656, so move it all to Red, and when that comes up you will have $93,312, almost enough for your Maserati. The chances of this happening, on a roulette wheel with a single zero, are 1/37×1/37×1/37×18/37 = 1 in 104,120.

So roulette easily gives the best chance of getting that shiny Maserati for a dollar. But given the odds involved, perhaps it’s best to stick there and start saving instead.

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